Is composition identity?

Abstract

Say that some things compose something, if the latter is a whole, fusion, or mereological sum of the former. Then the thesis that composition is identity holds that the composition relation is a kind of identity relation, a plural cousin of singular identity. On this thesis, any things that compose a whole (taken together) are identical with the whole. This article argues that the thesis is incoherent. To do so, the article formulates the thesis in a plural language, a symbolic language that includes counterparts of plural constructions of natural languages, and shows that it implies that nothing has a proper part. Then the article argues that the thesis, as its proponents take it, is incoherent because they take it to imply or presuppose that some things have proper parts.

Notes

Acknowledgements

The work for this article was supported in part by a SSHRC Insight Grant [Grant No. 435-2014-0592], which is hereby gratefully acknowledged. I presented its previous versions in the Mereology and Identity Workshop and at Nihon University. I would like to thank G. Lando, M. Carrara, and T. Iida for the invitations and the audiences for their comments and discussions. I would also like to thank two anonymous referees for Synthese for helpful comments on the penultimate version, P. Hovda for discussions on topics of this article, and Y. El Gebali and E. Darnell for editorial assistance. The penultimate version of this article was written while I was visiting Hokkaido University as a visiting scholar in 2018. I am grateful to T. Yamada and K. Sano for their invitation and hospitality during the visit. Needless to say, I am solely responsible for any errors and infelicities in this article.

Appendix

After giving an argument that rests on the Dodge Thesis (see Sect. 2.5), Sider gives an “alternative argument from strong composition as identity to unrestricted composition” (2007: p. 61). This argument rests on five theses, three of which involve modality. Using ‘Exist’ as a plural language predicate for existence, we can formulate the theses as follows:

M1.

Πxs ◊ ∃yComp(xs, y).

M2.

□ Πxs ∀y (Comp(xs, y) → xs≈y).

M3.

□ Πxs ∀y [xs≈y → □ (Exist(xs) → Exist(y) ∧ xs≈y)].

M4.

Πxs ∀y (xs≈y → Comp(xs, y)).

M5.

ΠxsExist(xs).

Sider argues that these imply Unrestricted Composition (UC):

UC.

Πxs ∃yComp(xs, y).

I think there are two major problems with this argument. One of them is specific to the argument. The other is an instance of the general problem with arguing that proponents of CAI must accept UC by deriving UC from CAI together with additional theses (see the last paragraph of Sect. 2.5).

(M1)–(M5) do not imply UC, for M1 does not mean that it is possible for any (actually existing) things to compose something while they all exist together. Unpacking the thesis using Definitions 1 and 2 (Sect. 2.1) yields M1ʹ:

M1ʹ.

Πxs ◊ ∃y [∀z(zΗxs → z < y) ∧ ∀z(z < y → ∃w(wΗxs ∧ wOz))].

Here the singular quantifier phrases (e.g., ‘∀z’) occur in the scope of the possibility operator and are restricted, for each possible world, to the things that exist in that possible world. Thus M1 does not require that there be a world in which something is composed by, e.g., all the things that actually exist, but only that there be a world with something composed by all those among them that exist in that world. This is satisfied if there is a world that has just one of the many actual existents and where the one object composes itself. Now, consider a two-world model satisfying ‘□ ∀x x < x’ that includes (a) the actual world whose only existents are two atomic objects and (b) the world whose only existent is one of those atomic objects. This model does not satisfy UC but satisfies all of M1–M5.

Defenders of Sider might propose a modification of the argument. One might add additional theses or replace M1 with a thesis that Sider might have in mind:

M1*.

Πxs ◊ (Exist*(xs) ∧ ∃yComp(xs, y)),

where ‘Exist*’ is an undefined plural language predicate amounting to the English ‘exist together’ used above.94 But modifying the argument to reach a valid argument has a general problem noted above. M2 (a modal cousin of CAI-1) and M4 (i.e., CAI-2) imply CAI. So if M2 and M4 together with some additional theses (e.g., M1*, M3, M5) imply UC, this means that all of them (taken together) imply Monism: there is only one thing (in the actual world). So some of them must be rejected by those who hold CAI without accepting Monism.

Now, it might be useful to consider how M1* and M2–M5 imply Monism. Consider all the things that exist in the actual world (i.e., < x: x = x >). They exist together (in the actual world) by M5. And by M1*, there is a possible world, Wʹ, in which they all exist together and compose something, A. So all of them (taken together) are identical with A (in Wʹ) by M2. By M3 (and M5), A exists and is identical with them in the actual world. So any one of them must be identical with A in the actual word (PL3). This implies Monism.95 The culprits in this case are M1* and M2. Thus proponents of CAI who reject Monism must reject M1*.

Sider argues that M1 “is harder than one might think for defenders of strong composition as identity to deny” (ibid.: p. 61). I agree, but for a different reason: M1 (which is much weaker than M1*) does not imply that there is any possible object with a proper part. But the same does not hold for M1*, which I think Sider has in mind in talking of M1. M2 is a consequence of the view that the composition relation is literally the plural identity relation. Thus those who hold this view cannot accept M1* without committing themselves to Monism.96 Sider argues that “the defender” of CAI must accept M1* because he or she “identifies the people with the subatomic particles” in their bodies.97 I agree that usual proponents of CAI hold that people (or their bodies) have proper parts. But one cannot consistently combine this view with CAI, for CAI implies the Part-whole Triviality Thesis as we have seen (PL6 in Sect. 2.3).